There are three players with three alternatives A,B, and C. Players simultaneously vote and majority wins. If no majority then A wins. Payoffs are:
$U_1(A) = U_2(B) = U_3(C) = 2$
$U_1(B) = U_2(C) = U_3(A) = 1$
$U_1(C) = U_2(A) = U_3(B) = 0$
The example states (A,B,C), (A,A,A), and (A,B,A) are all NE. I get why these are NE (I can reason this out) but I do not understand the process to get here. I could write out all possible strategy profiles and try to reason each one out but there has to be a better and efficient way. I also cannot draw the matrix for this game as I usually would to solve for NE to find best response. The preferences here also violate transitivity, which is just adding to my confusion.
My question is how would I go about finding all NE?
Are there more NE than these 3 the example mentions?
How to find multiple strategy profiles that support these NE i.e I see A can be supported by (A,B,C) and (A,A,A) (if there are other NE).
This is Example 1.5 in Fudenburg and Tirole.
Edit: Ok I think (B,B,B) and (C,C,C) and (A,C,C) are the other NE. I can't find anymore. Is that correct?